Near-optimal inversion of incoherent scatter radar measurements: coding schemes, processing techniques, and experiments

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NEAR-OPTIMAL INVERSION OF INCOHERENT SCATTER RADAR MEASUREMENTS: CODING SCHEMES, PROCESSING TECHNIQUES, AND

EXPERIMENTS

BY

ROMINA NIKOUKAR

DISSERTATION

Submitted in partial fulﬁllment of the requirements

for the degree of Doctor of Philosophy in Electrical and Computer Engineering in the Graduate College of the

University of Illinois at Urbana-Champaign, 2010

Urbana, Illinois Doctoral Committee:

Associate Professor Farzad Kamalabadi, Chair Professor Erhan Kudeki

Professor Steven J. Franke Associate Professor Minh N. Do

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ABSTRACT

Accurate and efficient estimation of the key ionospheric state parameters such as electron density, ion composition, and electron and ion temperatures is required to understand fundamental issues of terrestrial plasma physics such as redistribution of energy and momentum, and coupling within terrestrial upper atmosphere regions. This work focuses on developing a modern coding scheme and inversion methodology using Arecibo incoherent scatter radar (ISR) to achieve efficient, and near-optimal estimates of the key ionospheric parameters. In particular, this work considers two aspects of the ISR inversion problem: (i) ISR lag estimates at individual altitudes, and (ii) modulation techniques that can provide more accurate estimates with a specific range resolution. These two aspects suggest a unifying framework for ISR inversion in which modern computational technology and ISR methodology are utilized in a robust estimation procedure.

The first contribution of this work is the development of a discrete forward model for F-region incoherent scatter measurements, where long pulses are utilized in transmission. The range smear-ing imposed on measurements by long pulses is modeled as a one-dimensional convolution along the range in the simplified case where the receiver sampling is instantaneous. The next major phase of this research is to develop an efficient hybrid technique that allows for estimation of plasma parameters by removing range smearing from measurements. The inversion technique incorporates both quadratic and edge-preserving regularization approaches in order to provide smooth plasma auto-correlation function (ACF) lag profiles in the presence of noise while still resolving the sharp gradients.

Another contribution of this thesis is to develop a technique for optimal modulation design in ISR experiments. The optimal resolution supported by ISR measurements is used as one criterion for the optimal design. The model order selection framework is applied to the problem at hand to ﬁnd the optimal resolution. The results indicate that, compared with a long pulse,

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amplitude-modulated codes yield finer range resolutions with nearly similar parameter estimation errors or smaller estimation errors with the same range resolution. In medium to high SNR scenarios, a smaller on-off ratio of the transmitted waveform is recognized as a determining factor for allowing more freedom in removing range ambiguities as well as resulting in improved statistical accuracy for integrated data in range and lag directions. In order to find the optimal amplitude modulation for Arecibo ISR measurements in medium to high SNR scenarios, a modified form of the sequential backward selection algorithm is applied to the space of all amplitude modulated pulses with a certain on-off ratio. Due to the vast search space, there is no possibility for an exhaustive search. Therefore, the problem of finding the optimal amplitude modulation is viewed as an optimization problem. Three optimality criteria, namely, sum of squared errors, uniformity of estimation errors, and condition number of convolution matrices, are considered.

The final contribution of this work was implementing and conducting several experiments in April 2004, August 2005, and July 2006 using the incoherent scatter radar at Arecibo Observatory, and applying the inversion technique to estimate the plasma parameters. In these experiments, the original mode of MRACF was modified to utilize amplitude modulation. The results of all these experiments verify that when the SNR is sufficiently high, compared with an unmodulated long pulse, improved range resolution with nearly the same statistical accuracy is obtained when an amplitude modulation is utilized.

The results of the developed methodology and experimental design of this work can be extended to other incoherent scatter radars (such as Jicamarca radar in Peru, and advanced modular incoherent scatter radar in Alaska) to improve the estimation task in other altitude and latitude regions, and to extract many further ionospheric parameters such as electric ﬁeld strength, conductivity, current, and neutral wind speed.

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To my father, my mother, and my sisters, and

my husband, Hossein, for their love and support

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ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my adviser, Dr. Farzad Kamalabadi, for his guidance, understanding, and patience during my graduate studies at the University of Illinois. He encouraged me to grow not only as a scientist but also as an instructor and an independent thinker. I am also thankful to him for encouraging the use of correct grammar and consistent notation in my writings and commenting on countless revisions of this manuscript. My special thanks goes to Dr. Erhan Kudeki for getting my graduate research started on the right foot, and always being there to listen and give advice. I am deeply grateful to him for the long discussions that helped me sort out the technical details of my work. I am also thankful to Dr. Steven Franke and Dr. Minh Do for commenting on my ideas and helping me understand and enrich them.

I am deeply indebted to Dr. Michael Sulzer, Dr. Nestor Aponte, and Dr. Sixto Gonzalez for their knowledge and advice. Their valuable comments and directions not only helped me in my research, but also taught me precious lessons for my future academic career. I also thank the staﬀ at the Arecibo Observatory for their support during my data collection.

Most importantly, none of this would have been possible without the incredible support of my parents, sisters, and husband. My family to whom this dissertation is dedicated, has been a constant source of love, concern, support, and strength all these years. My special thanks goes to my parents who made it possible for me to pursue and complete my PhD degree. I am also thankful to all my friends at the University of Illinois for their friendship and support.

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CHAPTER 1 INTRODUCTION

1.1 Motivation

Incoherent scatter radar (ISR) is the most powerful ground-based technique for studying the Earth’s ionosphere. The incoherent scatter (IS) echo is the result of the scattering of electromagnetic energy, radiated from radar, by electrons in the ionospheric plasma, which are themselves controlled by much slower, massive positive ions. The frequency spectrum of the received signal provides information about electron and ion temperatures, electron density, ion composition and velocity. The analytical relationship between the spectrum and these parameters has been well established in the literature, e.g. [Dougherty and Farely, 1960, 1961; Farley, 1966, 1969; Hagfors, 1961, 1971]. With the extraction of these parameters from incoherent scatter measurements, one can deduce many further ionospheric parameters such as ion composition, electric ﬁeld strength, conductivity and current, neutral air temperature and wind speed.

Although the exact forward theory of incoherent scatter was established more than four decades ago, inversion, the estimation of parameters from incoherent scatter spectra, has remained an open-ended problem, due to two major factors. The first complication stems from the fact that variation of different plasma parameters may give rise to similar changes in the IS spectrum [Vallinkoski, 1988], e.g. the distinction between changes of the spectra due to ion composition or temperature ratio is very difficult. The same is true for ion mass and ion temperature for fixed temperature ratio. This implies that spectrum, or equivalently its Fourier transform, i.e. auto-correlation function (ACF), needs to be retrieved as accurately as possible, since small errors in ACF may yield large errors in estimated parameters. The second factor is the range smearing of information from one altitude over a number of altitudes, which is due to the length of the transmitted pulse.

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The transmitted pulse should be long enough to provide suﬃcient spectral resolution in measuring the spectrum, or equivalently, to permit ionospheric ACF to be measured with suﬃcient lag extent. On the other hand, at any given time, the received signal includes contributions from a volume which extends 𝑐𝑇₂ in the radar line-of-sight direction, where 𝑐 is the speed of light and 𝑇 is the length of the pulse.

An example of the effect of range smearing is shown in Figure 1.1 where different lags of the ACF are plotted prior to and after imposing the range ambiguity for an un-coded modulation of length 280 𝜇s with a 430 MHz radar (thin and thick curves, respectively). The estimation of ionospheric parameters from the range-smeared ACF would result in a greater ion temperature than electron temperature, which is not physical [Nikoukar et al., 2008]. Conventional processing techniques utilize various ways to tackle the inclusion of range smearing into analysis. Range-gate analysis, for example, is based on the assumption of constant plasma parameters at each range gate (distance covered by the pulse length) and compensates for the effect of range smearing by a simple triangular weighting. This method, although simple and fast, is based on an unrealistic assumption and suffers from the coarse resolution of estimated parameters. Full-profile analysis, which attempts to estimate ionospheric parameters at all altitudes simultaneously, on the other hand, considers a full model for range smearing. This technique, however, suffers from its significant computational cost, which can be mitigated by coarser interpolation of the parameter space, hence sacrificing estimation accuracy. The limitations of these processing techniques have motivated the search for more robust approaches to the problem of ISR inversion that meet the challenges of modern ISRs. The goal of this research is to develop a modern ISR parameter estimation methodology to achieve efficient, near-optimal estimates of ionospheric parameters with a fine resolution. Primarily, this research will focus on the F-region ionosphere over the Arecibo Observatory, where the effect of molecular ions and collision can be simply neglected in the analysis. Moreover, the beam direction over Arecibo can be far from perpendicular to the magnetic field lines, where the incoherent scatter process is well known. This work will consider two aspects of the ISR inversion problem: (i) ISR lag estimate at individual altitudes and (ii) modulation techniques that can provide more accurate estimates with a specific range resolution. These two aspects suggest a unifying framework for ISR inversion in which modern computational technology and ISR methodology are utilized in a robust estimation procedure.

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0 50 100 150 200 250 300 0

0.5 1

Time lag ( sec)

Normalized ACF

at a given height

Figure 1.1 Normalized theoretical and measured ACF for a long un-coded pulse transmission and a constant ionosphere (thin and thick curves, respectively). Range smearing makes the shape of ACF so distorted that results of the ﬁt by standard nonlinear least-squares methods become poor. Notice that the above ACFs have been generated using a radar frequency of 430 MHz for a single ion ionosphere (oxygen). The lag spacing is 10 𝜇s and an un-coded long pulse of length 280 𝜇s is considered for transmission.

1.2 Context and research contribution

Since the development of ISR theory and construction of large incoherent scatter radars, such as the Arecibo radar, various methodologies were invented to deal with difficulties attributed to ISR parameter estimation. Generally speaking, they fall into two major categories of analysis techniques and coding schemes. To date numerous coding schemes have been primarily designed to make direct measurements of plasma ACF possible despite the pulse length. There are two main classes of coding techniques, amplitude modulation and phase modulation schemes, used for transmission. Amplitude modulation schemes such as double-pulse and multi-pulse techniques, pioneered in [Farley, 1969, 1972], exploit short pulses to obtain the spectra at individual altitudes. They, however, suffer from great sensitivity to the background and receiver noise. These techniques are suitable only for parameter estimation around the F-region peak where the back-scattered signal is rather high. Phase modulation schemes, such as alternating codes, maintain the fine spatial resolution of measurements, while exploiting the full duty cycle of radar, and are not as sensitive to the noise level. They, however, are only appropriate for low signal-to-noise ratio (SNR) scenarios,1

where the lag estimate errors are independent from each other. As a result, in the case of the Arecibo radar, where the transmitted power is high and the transmitting and receiving antenna area is large, this technique loses its appeal over other existing coding techniques.

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Currently, the usual mode of operation for this type of measurements at Arecibo Observatory is the simple long-pulse technique. The major difficulty associated with the long-pulse modulation is that it can only provide range-smeared (indirect) measurements of plasma ACF. The current analysis techniques developed to deal with this range smearing of information suffer from severe sensitivity to noise, coarse spatial resolution, or significant computational expense. Although the relationship between the measured signal ACF and the plasma ACF at individual altitudes has been thoroughly derived in the work of Lehtinen and Huuskonen [1996], to date no systematic approach exists to exploit this relationship in favor of more efficient methodologies. In this work, we present an efficient, near-optimal approach to the problem of incoherent scatter radar inversion which is extremely robust in the presence of system and background noise. The key to this new method is to consider the system of measurements as a combination of two linear and nonlinear systems. We formulate the estimation of plasma parameters as the result of linear inversion of measured signal ACF (to achieve plasma ACF at individual altitudes) and then of nonlinear inversion of plasma ACF (to obtain plasma parameters). Although the proposed analysis approach attempts to obtain estimates of plasma ACF at individual altitudes (separated by the distance equal to lag resolution), it is only able to partially remove the smearing; and thus there is a limit to the resolution of the estimated physical parameters.

Another drawback associated with un-coded long-pulse measurements is that they suﬀer from high correlation between ACF lag estimate errors. This correlation plays an important role in ISR error analysis, that is, determining the uncertainty level in estimated parameters. First, in order to avoid underestimation of error bars, one has to account for the full error covariance matrix.2 _{In addition, when SNR is high, random ﬂuctuations in the ACF estimates originate}

in part from the self-noise, which is generated by the uncorrelated scatter from the transmitted signal from all simultaneously illuminated altitudes [Lehtinen and Huuskonen, 1996]. When the self-noise is dominating, the accuracy of observations cannot be increased by further increasing the signal strength because the random error in data increases at the same rate. Another eﬀect of the correlation between lag estimate errors is related to the range integration. Because the estimates are highly correlated in high SNR regimes, the addition of 𝑁 ACFs from adjacent altitudes does

2_{An error covariance matrix is formed from the correlation between errors of diﬀerent lag estimates received at}

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not yield an √𝑁-fold increase in estimation accuracy as would be the case for fully independent

observations.

In this work we present amplitude modulation (AM) as the solution to reducing the correlation between ISR lag estimate errors. This reduction in correlation, in turn, allows for improving range resolution especially in high SNR regimes. For this purpose, we study and compare the performance of amplitude modulation with that of long-pulse modulation in terms of statistical accuracy and range resolution by means of both numerical simulations and real data analysis. We show that the correlation between lag estimate errors is highly aﬀected by the envelop of the modulated waveform for a given SNR, and that amplitude modulation reduces this correlation signiﬁcantly. We show that this lower correlation results in improved accuracy when measurements are integrated in range and lag. Using model order selection methods, we also derive a mathematical measure for estimating the fundamental range resolution supported by ISR measurements.

Moreover, we present parameter estimation results of several F-region experiments that we con-ducted with Arecibo radar in 2004, 2005, and 2006. In these experiments, long-pulse and various amplitude modulation codes were alternated in transmission to make a direct comparison between the estimation results possible. The on-off ratio of the transmitted waveform is represented as a major determining factor in the performance of an amplitude modulated code. However, slight dif-ferences in performances of AM pulses with the same on-off ratio necessitate consideration of other related factors, such as uniformity of the number of integrated heights for different lags, or condi-tion number of convolucondi-tion matrices. In this work, we present close-to-optimal waveform design for ISR experiments using subset selection methods. Our method, however, provides preliminary results and its convergence properties have yet to be developed.

The developed ISR methodology can also be extended to other incoherent scatter radars such as Jicamarca radar in Peru or advanced modular incoherent scatter radar (AMISR) in Poker Flat, Alaska. Although the requirements of the plasma correlation measurements for these radars are quite diﬀerent from those for the Arecibo radar, the developed methodology can be expanded to meet the requirements.

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1.3 Dissertation overview

This dissertation is organized, as follows: Chapter 2 presents an overview of the incoherent scatter theory, the incoherent scatter radar equation, and the concepts of range smearing and ambiguity. It also covers the conventional coding schemes and conventional incoherent scatter data process-ing techniques. The first contribution of this work is the development of a hybrid technique that allows for efficient estimation of plasma parameters from ISR measurements. In Chapter 3, a tech-nique is developed which combines regularization theory and deconvolution methods. Two different regularization methods, quadratic and edge-preserving methods, are considered. A statistical in-terpretation of the proposed algorithm is given using statistical formulation. Numerical results are presented.

Another contribution, presented in Chapters 4, 5, and 6, is the development of a framework for determining the fundamental resolution, and optimal experiment design, in ISR measurements. We also compare the performances of long-pulse and amplitude modulation, which results in lower correlation in lag estimate errors. Numerical estimates of ACF lag variance reduction are presented for a long pulse and several amplitude modulated pulses when measurements are integrated in range and lag directions. Moreover, the improvement in range resolution for estimated parameters is presented for AM pulses using numerical simulation in Chapter 4. Chapter 5 describes several ISR experiments conducted using the radar at Arecibo observatory, and presents a compelling comparison between the performances of long pulse and AM pulses with different on-off ratios, as well as AM pulses with the same on-off ratio. In Chapter 6, optimal design of the transmitted waveform in F-region ISR experiments is considered, and a practical algorithm is proposed for finding the optimal pulse configuration.

Finally, Chapter 7 provides concluding remarks, summarizes the dissertation’s contributions, and discusses some future research directions.

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CHAPTER 2 PRELIMINARIES

2.1 The basics of incoherent scattering of radio waves

The idea of incoherent scatter radars was initiated by W. Gordon in 1958. He suggested that if a powerful beam of radio waves with a frequency well above the plasma frequency was sent vertically through the ionosphere, an extremely small but still measurable amount of power would scatter back to the ground by small-scale random ﬂuctuation of electrons in the ionosphere. The backscattered signal could be roughly estimated as the sum of the powers scattered from each electron, as given by the Thompson electron cross-section [Kudeki, 2003; Nygren, 1996]. Since the electrons are randomly distributed in the ionosphere, they scatter incoherently, hence the name “incoherent scatter,” and the power scattered by each is summed to obtain the total signal. Moreover, owing to the thermal motions of the electrons, the scattered radiation would be subject to a Doppler broadening, with a frequency deviation of the order of (𝐾𝑇 /𝑚𝑒)12/𝜆₀, where 𝐾 is the Boltzmann constant, 𝑇 is the

temperature, 𝑚𝑒 is the electron mass, and 𝜆0 is the radar wavelength.

The ﬁrst successful ISR experiment was performed by Bowles in 1958 using extremely high-powered equipment. His results indicated that the returned power was of the order of the power predicted by Gordon; however, the observed Doppler broadening was much smaller than what Gordon had anticipated. Diﬀerent groups of scientists investigated this discrepancy [Dougherty and

Farely, 1960, 1961; Farley, 1966; Hagfors, 1961, 1971]; and although they used diﬀerent theoretical

approaches, they all arrived at the following conclusion: Provided that the Debye length (the distance over which the influence of the electric field of an individual charged particle is felt by the other charged particles inside the plasma [Bittencourt, 2004]) is sufficiently large, the ions play an important role in determining the Doppler spread, although it is the electrons that do the actual

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scattering and that the eﬀect of ions in the form of Coulomb collisions must be considered in the electron scattering.

In the next section, we describe the basic idea behind the incoherent scatter of the radio waves with pulsed incoherent scatter radars, and elaborate on the expression of the incoherent scatter spectrum, or equivalently its Fourier transform, auto-correlation function, as a function of iono-spheric parameters. We then demonstrate the eﬀect of varying parameters on the resulting spectra or ACFs by a few examples.

2.1.1 Variation of the spectrum as a function of parameters

In the presence of a magnetic ﬁeld (B) in a plasma with Coulomb interactions and a beam direction away from perpendicular to B, the spectrum of the electron density ﬂuctuation, < ∣𝑛𝑒(k, 𝜔)∣2 >,

can be represented by < ∣𝑛𝑒(k, 𝜔)∣2 > = 2𝑁0 ∣𝑗(𝑘2_ℎ2 𝑒+ 𝜇) + 𝜇𝜃𝑖𝐽(𝜃𝑖)∣2 𝑅𝑒{𝐽(𝜃_𝑘√_2𝐶𝑒_𝑒)} + ∣𝑗 + 𝜃𝑒𝐽(𝜃𝑒)∣2 𝑅𝑒{𝐽(𝜃_𝑘√_2𝐶𝑖_𝑖)} ∣𝑗(𝑘2_ℎ2 𝑒+ 1 + 𝜇) + 𝜃𝑒𝐽(𝜃𝑒) + 𝜇𝜃𝑖𝐽(𝜃𝑖)∣2 (2.1)

where < . > denotes the expectation operation, 𝑁0 represents electron density, 𝜇 is the electron-ion

temperature ratio 𝑇𝑒/𝑇𝑖, ℎ𝑒 is the Debye length, and 𝜃𝑠 represents the normalized velocity of the

particle 𝑠. The Gordyeve-type integral 𝐽(𝜃𝑠) is described by

𝐽(𝜃𝑠) =

∫ _∞

0 𝑑𝑡𝑒

−𝑗𝜃𝑡_𝑒−[𝑡24 sin2𝛼+_𝜙2𝑠1 sin2(𝜙𝑠𝑡2 )cos2𝛼] _(2.2)

where 𝛼 is the magnetic aspect angle (the complement of the angle between propagation vector k and magnetic ﬁled B), and 𝜙 denotes the normalized gyrofrequency of the species 𝑠 (Ω𝑠), i.e.

𝜙𝑠= _𝑘√Ω_2𝐶𝑠 _𝑠 (see [Kudeki, 2003] for a complete discussion).

In thermal equilibrium where 𝑇𝑒= 𝑇𝑖, the spectrum consists of two peaks close to the origin with

widths corresponding to ion thermal velocities. Therefore, the spectrum of a plasma has a small dip at the central frequency corresponding to no Doppler shift. In order to provide a qualitative picture of the eﬀect of various parameters on the spectrum (and its Fourier transform equivalent, ACF), spectra (and ACFs) at various parameter combinations are plotted in Figure 2.1 (a) and (b). Plots

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−200 −10 0 10 20 0.2 0.4 0.6 0.8 1 Frequency (kHz) −200 −10 0 10 20 0.2 0.4 0.6 0.8 1 Frequency (kHz) 0 100 200 300 400 500 −0.5 0 0.5 1 Lag (µ s) T i = 500 K T i = 1000 K T i = 1500 K T i = 2000 K 0 100 200 300 400 500 −0.5 0 0.5 1 Lag (µ s) µ = 1 µ = 1.5 µ = 2 µ = 2.5

Figure 2.1 Variation of the ISR spectrum and ACF as a function of ion temperature (left panels) and temperature ratio (right panels), respectively.

in the left panels show the variation of the spectrum and ACF as a function for ﬁxed temperature ratio (𝜇 = 2) and as a function of ion temperature, whereas plots in the right panels show the variation of the spectrum and ACF for ﬁxed ion temperature (𝑇𝑖 = 1000 K) and as a function of

temperature ratio. All plots have been generated for frequency of 430 MHz and a 100% oxygen ionosphere. As seen from the plots, for a ﬁxed 𝜇, the ion line becomes broadened as ion temperature increases. Moreover, the peak-to-valley ratio also increases with increasing temperature ratio with ion temperature ﬁxed.

In the next section, we describe how we can infer the plasma spectrum or ACF from an incoherent scatter radar.

2.2 Principles of pulsed radar operations, the soft-target radar

equation, and the ambiguity function

In order to introduce the soft-target radar equation, we ﬁrst provide a quick overview of the principles of pulsed radar ISR operation. For this purpose, let us denote the envelope and the length

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T Sampling

Time range

r=c*t /2

t seconds

Figure 2.2 Transmission and reception scenario in a typical incoherent scatter radar. A pulse (modulated or unmodulated) of length 𝑇 is transmitted through the ionosphere, and the return is sampled at 𝛿𝑡 intervals.

of the transmitted waveform in the radar experiment by 𝑉0(𝑡) and 𝑇 , respectively. Suppose the

pulse is a baseband waveform modulated by a sinusoidal wave of frequency 𝜔0. The corresponding

wavelength and wave vector will be 𝜆0 = 2𝜋𝑐_𝜔₀ and k0 = 2𝜋_𝜆₀ˆk, respectively, where 𝑐 is the speed of

light and ˆk is the direction of wave transmission. The returned signal is then sampled with a period

𝛿𝑡 (Figure 2.2) [Nikoukar et al., 2008].

The scattered signal from the electron density ﬂuctuations in the ionosphere can be represented by [Kudeki, 2003] 𝑣(𝑡) ∝ ∫ r𝑛(r, 𝑡)𝑒 −𝑗2𝑘0𝑟_𝑉 0(𝑡 −2𝑟_𝑐 )𝑑r (2.3)

where 𝑛(r, 𝑡) represents the ﬂuctuating component of the plasma electron density at range r = 𝑟ˆr along the direction ˆr. The proportionality factor, 𝑍(r), describes all geometrical eﬀects and can be written as:

𝑍(r) = 𝑅𝑟𝑎𝑑𝐺(ˆr)𝑟_2𝑘 𝑒

0𝑟2 (2.4)

where 𝑟𝑒 ≈ 2.8×10−15m is the classic electron radius, 𝑅𝑟𝑎𝑑 represents the radiation resistance of the

antenna, and 𝐺(ˆr) is the antenna gain along the direction ˆr. Note that the received signal spectrum is centered at the carrier frequency, 𝜔0, instead of zero, as the antenna output is modulated by the

carrier signal. At the coherent detector, the input is mixed with the signal derived from a local oscillator centered at the carrier frequency and is low-pass ﬁltered. This task brings the output signal to the baseband again, centered at zero frequency. The exponential term in (2.3) represents

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this mixing operation.

Since there are many scattering electrons in the ionosphere, the scattered signals (or received voltages) can be treated as Gaussian random variables to a very good approximation, according to the central limit theorem [Papoulis, 1986]. Therefore, instead of working directly with voltage sam-ples, we model their joint statistics in the form of the auto-correlation function, < 𝑣(𝑡)𝑣(𝑡 + 𝜏) >, where 𝜏 represents the time lag and (.) denotes the conjugation operation, respectively.

From (2.3), the ACF of the received voltage can be expressed as

< 𝑣(𝑡)𝑣(𝑡 + 𝜏) > ∝ ∫ r′ ∫ r < 𝑛(r, 𝑡)𝑛(r ′_{, 𝑡 + 𝜏) > 𝑉}₀_{(𝑡 −}2𝑟 𝑐 )𝑉0(𝑡 + 𝜏 − 2𝑟′ 𝑐 )𝑒−2𝑗𝑘0(𝑟 ′_−𝑟) 𝑑r𝑑r′ (2.5)

where r′ _{= 𝑟}′ˆr′ _{denotes the range 𝑟}′ _{along the direction ˆr}′_{. Within the integrand above <}

𝑛(r, 𝑡)𝑛(r′_{, 𝑡 + 𝜏) > is the space-time ACF of the density ﬂuctuation 𝑛(r, 𝑡). With the}

assump-tion that ﬂuctuaassump-tions have homogenous and staassump-tionary statistics which vanish rapidly with an increasing magnitude of x ≡ r′_{− r, we can proceed as}

< 𝑣(𝑡)𝑣(𝑡 + 𝜏) > ∝ ∫ r𝑉0(𝑡 − 2𝑟 𝑐 )𝑉0(𝑡 + 𝜏 − 2𝑟 𝑐 )𝑑r ∫ x< 𝑛(r, 𝑡)𝑛(x + r, 𝑡 + 𝜏) > 𝑒 −2𝑗k0.x_𝑑x (2.6)

The inner integral represents a spatial Fourier transform, denoting the ACF of electron density ﬂuctuations at altitude 𝑟, time lag 𝜏, and the wave vector (k = −2k0), as follows:

𝑅(k, 𝑟, 𝜏) ≡

∫

x< 𝑛(r, 𝑡)𝑛(x + r, 𝑡 + 𝜏) > 𝑒

−2𝑗k0.x_𝑑x _(2.7)

Notice that this ACF of electron density ﬂuctuations is the same as the plasma ACF whose an-alytical expression (as a function of radar and plasma parameters) was already introduced in the previous section.

By inserting (2.7) into (2.6) we obtain [Kudeki, 2003]:

< 𝑣(𝑡)𝑣(𝑡 + 𝜏) > ∝ ∫ r𝑉0(𝑡 − 2𝑟 𝑐 )𝑉0(𝑡 + 𝜏 − 2𝑟 𝑐 )𝑅(k, r, 𝜏)𝑑r (2.8)

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The above equation which demonstrates the relationship between the received voltage and the target in a statistical sense is called the soft-target radar equation and provides the necessary background for estimation of ionospheric parameters. Inspection of this equation brings us to the following two observations:

1. The incoherent scatter radar equation holds for each lag of ionospheric ACF independently from other time lags. Hence, to evaluate the plasma ACF at time lag 𝜏, we need to consider only the same lag of the received signal ACF.

2. Using incoherent scatter radar, it is not possible to achieve point estimations of the plasma ACF. Instead, weighted averages of this quantity over a ﬁnite-range interval are obtained. The weights are merely dependent on the modulated waveform and vary from lag to lag. The functions describing the averaging operation on the underlying plasma ACF are called soft-target radar ambiguity functions, (p𝜏(𝑡) = 𝑉0(𝑡)𝑉0(𝑡 + 𝜏)). These functions essentially indicate

that the signal coming from range 𝑟 contains information from several altitudes, where the altitude interval is equal to the distance covered by the product of the waveform and its shifted version. Note that the zeroth lag contains the most range smearing of information. As we move to farther lags, this altitude interval decays as the common part between the pulse and its shifted version diminishes.

Notice that in a more general case, the ambiguity functions are dependent on both range vari-ables and time lag, as developed in Lehtinen and Huuskonen [1996] and Holt et al. [1992]. This dependence on time lag is caused by a non-ideal receiver, whose impulse response contains the time average of previous samples. For now, however, we assume the receiver has a suﬃciently narrow impulse response, and as such the ambiguity in the lag direction is negligible. The expansion of the radar ambiguity function to the 2-dimensional case is considered in Section 3.3.

Also note that our definition of the soft-target ambiguity function differs from that of hard-target radar applications where it is defined as (see for example [Blahut, 2004])

𝜒(𝜏, 𝜈) =

∫

𝑉0(𝑡)𝑉0(𝑡 + 𝜏)𝑒−𝑗2𝜋𝜈𝑡𝑑𝑡 (2.9)

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variable, 𝜏, and doppler resolution, 𝜈. The hard-target radar equation is used as a measure of detectability of two hard targets with separation in range and velocity. The difference between the two definitions is due to different filters being used in radar receivers. In a hard-target application, normally a matched filter is used, whereas in soft-target radars, a receiver with a boxcar impulse response of short length is often utilized. Through the rest of this work, we use the term “ambiguity function” to refer to soft-target radar applications only.

2.3 Conventional coding schemes in incoherent scatter

experiments

In this section, we describe the conventional coding schemes that are used in typical F-region incoherent scatter experiments. These modulation techniques include long-pulse and multi-pulse techniques, and alternating codes.

2.3.1 Long-pulse technique

In long-pulse transmission, let 𝑉0(𝑡) = 1 for0 ≤ 𝑡 < 𝑇 , where 𝑇 represents the length of the pulse.

In this case, the one-dimensional ambiguity function 𝑝𝜏(𝑡) can be represented as follows:

𝑝𝜏(𝑡) = ⎧  ⎨  ⎩ 1 if 0 ≤ 𝑡 < 𝑇 − 𝜏 0 otherwise

In other words, the ambiguity function is a square pulse whose width decreases as the time lag increases. Therefore, in long-pulse transmission, it is not possible to retrieve point estimations of the plasma ACF directly from the data. To do so, one needs to account for the range ambiguity and has to devise methods that are described in the next section as well as the next chapter.

2.3.2 Multi-pulse technique

The class of multi-pulse codes pioneered by Farley [1972] is a set of amplitude modulated pulses, which can achieve point estimation of ionospheric ACF (nonzero) lag estimates at individual

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alti-tudes without ambiguity. A classical pulse-code modulation meets the following conditions: 1. The code consists of 𝑁 pulses of equal duration (bit length).

2. The distance of any pair of pulses is an integer multiple of the smallest (basic) inter-pulse distance (𝜏), and all distances are diﬀerent. The lag resolution is set to be equal to the basic inter-pulse distance.

Although the multi-pulse technique can provide the ionospheric ACF estimation at individual altitudes, its major drawback is its low duty cycle, and thus its high sensitivity to the background noise level. For example, a six-pulse code with a bit length of 17 𝜇s, will have a total length of 306 𝜇s (see Figure 2.3). This configuration yields a duty cycle of only 33%. As the number of short pulses increases, the duty cycle decreases. For example, a 7-pulse code has a duty cycle of 26% only. In order to facilitate the full radar duty cycle, and therefore to reduce the sensitivity of the method, the technique is usually implemented in the form of interlaced codes. In an experiment with these interlaced codes, the transmitted frequency is changed rapidly so that gaps are not left in the transmission. Another drawback of the multi-pulse codes is that the zero lag measured by this technique is normally discarded. The reason is that the shape of the ambiguity function for the zeroth lag differs from that of others, and its corresponding scattering volume consists of a number of individual volumes instead of a single volume. The use of multi-pulse zero lag data, however, can be used to improve incoherent scatter radar power profile accuracy, as suggested by Lehtinen

and Huuskonen [1986]. 0 50 100 150 200 250 300 −1 −0.5 0 0.5 1 1.5 2 time (µs) 6−pulse

Figure 2.3 Six-pulse technique. The baud is 17 𝜇s and the pulses are located at 0, 17, 136, 187, 221, and 289 𝜇s.

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2.3.3 Alternating codes

Alternating codes are a series of phase-coded pulses which are transmitted one at a time through the ionosphere, and once all the pulses are transmitted, the cycle starts over again. Each single pulse consists of a combination of elementary pulses (bauds) with signs ±1. These signs are changed from pulse to pulse in a way that the ambiguity function due to all of these pulses has a single peak on a zero background.

The working principle of this coding method is shown in Figure 2.4. The matrix on the left shows a set of four phase-coded pulses (scan count), where each pulse consists of four elementary pulses with varying signs. The matrix marked by 𝑊1 denotes the sign of the ambiguity functions

of the ﬁrst lag. This is obtained by multiplying each column of the matrix by its adjacent one. The matrices in the square region are obtained by further multiplying each column of 𝑊1 by the

first through third column. According to the figure, all but one of the the columns of each matrix sum to zero. Recalling that each column in the original envelope of the pulses corresponds to one particular altitude, one can easily see that the undesired signals from the other two altitudes have been canceled out, while the signal from the altitude of interest is preserved. The same argument holds for other lags of the ambiguity function, 𝑊2 and 𝑊3. The only difference is that as we

consider higher lags, the number of columns (altitudes) decreases, until a single column is left for

𝑊3.

In their work, Lehtinen and Huuskonen [1996] have presented an analytical solution to the prob-lem of choosing signs of every bit in all scans such that the whole set shows the desirable properties of single peaks for the ambiguity functions. For this purpose, they exploit one of the properties of the Hadamard matrices, which is orthogonality of rows [Harwit and Sloane, 1979]. As a result, certain arrangements of the rows or columns achieve the analytical cancelations. They have also shown that in situations where the received impulse response has a ﬁnite width, as opposed to a

𝛿-like response, the required number of scans in each cycle is twice the number of elementary pulses

in each scan. This requirement reduces the time resolution of the measurements by a factor of 1/𝑁𝑝, where 𝑁𝑝 is the number of scan counts in one complete cycle.

Although alternating codes guarantee the exact cancelation of the signal contribution from un-wanted regions, they fail to remove the covariances between the lag-estimate errors in an ISR

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lses W1

-Figure 2.4 Working principle of the alternating codes. Ambiguity functions are shown for all possible three lags (𝑊𝑖, where 𝑖 is the lag number). The matrices next to each ambiguity matrix

are formed by multiplying each column (altitude) of this matrix by other columns (altitudes). This multiplication results in the canceled contributions from these columns, while preserving the signal from the ﬁrst altitude.

experiment. Covariance calculations, however, require the lag profile matrices from all the codes, not only the final results. This requirement makes the method even more computationally expen-sive, but makes it more efficient when the SNR is very low and the lag estimate errors are nearly independent.

2.4 Current methodologies in incoherent scatter inversion

Let us assume that one of the above-mentioned coding schemes has been utilized in an ISR experi-ment, and estimates of the received signal ACF lags (or plasma ACF lags provided that multi-pulse or alternating codes are used) are available. In this section we provide an overview of the sta-tistical framework for inversion of incoherent scatter radar measurements to obtain estimates of ionospheric parameters. We formulate the problem of inversion in terms of the maximum likelihood (ML) principle. Next, we explain conventional inversion methodologies, height-by-height analysis and the full-proﬁle technique.

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function

g(a)

m (ACF or spectrum)

+

𝜖𝑚 (noise)

a(Te, Ti, Ne, p, ...) _{g: nonlinear incoherent scatter}

Figure 2.5 Incoherent scatter process as a hypothetical system, where 𝑎 which is a vector of plasma parameters is the input of the system, 𝑚, the plasma spectrum or ACF, is the output, and the nonlinear function of incoherent scatter, which relates the parameters to plasma spectrum, is denoted by 𝑔(.).

2.4.1 Statistical framework for incoherent scatter inversion

Let us consider the incoherent scatter process as a system, shown in Figure 2.5 and denoted by g(.), in which the ionospheric state parameters form the input, a, and the spectrum or ACF of the received signal forms the output, m. The relationship between the measurements and the desired parameters can be written as

m = 𝑔(a) + 𝜖 (2.10)

where 𝜖 denotes the noise of the system and in the most general case is signal dependent. However, it gains Gaussian characteristics as the measurements are integrated over many transmissions. Therefore, it can be well described by its mean, 𝜇 =< 𝜖 >, and covariance matrix ∑_𝜖 =< 𝜖𝜖𝑇 _>.

A nonzero mean would indicate biases, and without loss of generality can be considered as zero. The conditional probability density function now can be expressed by

𝑝(m∣a) = 1 (2𝜋)𝑛/2_∣Σ_𝜖_∣12 exp ( −1₂(m − 𝑔(a)𝑇Σ−1_𝜖 (m − 𝑔(a))) ) (2.11)

where 𝑛 is the number of data points. The joint probability density function of m and a can be described as 𝑝(m, a) = 𝑝(a)𝑝(m∣a) ∼ 𝑝(a) 1 (2𝜋)𝑛/2_∣Σ_𝜖_∣12 exp ( −1₂(m − 𝑔(a))𝑇Σ−1_𝜖 (m − 𝑔(a)) ) (2.12)

Let us now suppose that the a priori density of the parameters (𝑝(a)) is approximately constant in the regions where the conditional probability density function is signiﬁcantly diﬀerent from zero.

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The ML estimate is then equivalent to minimizing the quadratic form in the exponent in (2.12), i.e. 𝜒2 _{= −}1

2(m − 𝑔(a)𝑇Σ−1𝜖 (m − 𝑔(a))). Thus, under the assumption of Gaussian errors, the

ML estimate can be obtained via the quadratic or least-squares optimization procedures. If the matrix ∑_𝜖 is diagonal, this quadratic form is reduced to the sum of the diﬀerences between the components of the predicted data, 𝑔(a), and the components of the measurements weighted by the inverses of the diagonal elements of∑_𝜖. On the other hand, in situations where∑_𝜖 is not diagonal, the quadratic form of (2.12) deﬁnes a generalized least-squares estimation method.

Linear approximation

Let us suppose that the correct values of the parameters are a0 and the predictive measurements

are denoted by m0 = 𝑔(a0). If the errors are small, the solution to the least-squares problem is very

close to a0, and as such we can use the Taylor expansion of the ﬁrst degree to state the nonlinear

quadratic form of (2.12)

m − m0 = 𝑔(a) − 𝑔(a0) + 𝜖 = A(a − a0) + 𝜖 (2.13)

where A𝑖𝑗 = ∂g𝑖/∂a𝑗 is the partial derivative of the incoherent scatter function with respect to

plasma parameters. Thus the errors derived are only valid in the limit of small ﬂuctuations in the measurements. In practice this condition can be satisﬁed by using longer integration times. Replacing a − a0 and m − m0 by a and m, respectively, yields

m = Aa + 𝜖 (2.14)

The linear formulation is necessary and useful because the likelihood or posteriori distribution, and thus the solution, can be expressed in analytical forms. Here, we focus on the ML estimate where, with the assumption of the Gaussian noise, the likelihood can be described as

𝑝(m∣a) ∼ exp

(

−1₂(m − Aa)𝑇Σ−1_𝜖 (m − Aa) )

(2.15)

In order to bring the expression for the transitional density to a standard quadratic form, we ﬁrst consider the Cholesky decomposition of the inverse noise covariance matrix as Σ−1

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∑

𝜖is positive semi-deﬁnite). We proceed by simpliﬁcation of (2.15) and using the following identity

B(BT_B)−1_BT _{= I} 𝑝(m∣a) ∼ exp ( −1₂(m − Aa)𝑇_D𝑇_{D(m − Aa)}) = exp (

−1₂(Dm − DAa)𝑇(DA[(DA)T(DA)]−1(DA)T)(Dm − DAa) ) = exp ( −1₂(A𝑇Σ−1_𝜖 m − A𝑇Σ−1_𝜖 Aa)𝑇(A𝑇Σ−1_𝜖 A)−1(A𝑇Σ−1_𝜖 m − A𝑇Σ−1_𝜖 Aa) ) (2.16)

Factoring the term Q = A𝑇_Σ−1

𝜖 A and further simpliﬁcation yields

𝑝(m∣a) ∼ exp ( −1₂(Q−1_A𝑇_Σ−1 𝜖 m − a)𝑇Q(Q−1A𝑇Σ−1𝜖 m − a) ) (2.17)

The ML estimate, which minimizes the likelihood, can be analytically expressed as

a𝑀𝐿 = Q−1AΣ−1𝜖 m (2.18)

where its error covariance matrix is given by

Q−1 =(A𝑇Σ−1_𝜖 A)−1 (2.19)

The matrix Q is called the Fisher information matrix [Poor, 1994]. Clearly, all the measurement components make certain contribution to the Fisher information matrix.

The statistical inversion as described above is applicable to ISR inversion only in situations where the plasma ACF is readily available, by utilizing multi-pulse or alternating codes, for example. In cases where the modulation does not provide ambiguity-free measurements, one has to account for the eﬀect of ambiguity in some way. Below, we review two of such methods, namely, height-by-height analysis and the full-proﬁle technique. In the next chapter, we introduce a hybrid technique for the inversion of ISR measurements.

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r1 r3 r2 T lower lags Higher lags Pulse length

Figure 2.6 Range-time diagram in a long-pulse transmission. The ﬁtting is performed on the data from altitudes 𝑟1, 𝑟2, and 𝑟3 only. The number of altitudes contributing to the signal from 𝑟1

decreases as we move to higher lags.

2.4.2 Height-by-height analysis

The height-by-height analysis has been traditionally used in the analysis of long-pulse modulation in which measurements suﬀer from range ambiguity. The method relies on the assumption that parameters do not vary over a range-gate (an altitude interval which, in the most common case, is equal to the distance covered by the transmitted pulse). One then ﬁts the plasma parameters by least-squares methods to the chosen measured ACF using the theory of incoherent scatter at individual altitudes with range-gate separation (altitudes 𝑟1, 𝑟2, and 𝑟3 in Figure 2.6 separated by

range-gates).

Prior to least-squares estimation, however, a number of modiﬁcations should be applied to the received signal ACF. These changes are required to compensate for the range smearing, and without them the shape of the ACF becomes so distorted that the results of least-squares ﬁtting become infeasible. One major reason is that the ACF of the received signal is the average of the ACF from several altitudes, and due to the nonlinear nature of the problem, the estimated parameters need not be close to the averages of the plasma parameters within the scattering volume.

Triangular weighting is a method which compensates the effect of range smearing and is based on weighing different lags of the measured ACF to make the effective weight of all lags equal. With a transmitted pulse of 𝑇 s, the zeroth lag of the received signal ACF is the superposition of signals coming from 𝑇 different altitudes separated by 1 s, whereas the (𝑇 − 1)th lag measured signal ACF results from the signal coming from only one altitude. Thus, in effect, the range smearing weighs the zeroth lag 𝑇 times higher than the (𝑇 − 1)th lag, and the task of triangular weighting is to

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undo this weighing process.

The hight-by-height analysis is also called range-gate analysis, as the data is processed at each range-gate separately. The technique, although simple and fast, suffers from the underlying un-realistic assumption along with the coarse resolution of estimated parameter profiles. Introduced biases in parameter profiles have also been reported in electron density in regions below the peak height [Lehtinen and Huuskonen, 1996; Holt et al., 1992].

2.4.3 Full-proﬁle analysis

In this section, we ﬁrst formalize a simple version of the idea of the full-proﬁle analysis technique. We next explain the issues regarding practical implementations of such methods.

As opposed to the hight-by-height technique where only a few parameters are estimated using a nonlinear optimization procedure, in the full-proﬁle method the unknown is a much longer vector giving the plasma variables at ionospheric-height grid points. This vector can be constructed from the elements of the following matrix

a = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑇𝑒(𝑟1) 𝑇𝑖(𝑟1) 𝑁𝑒(𝑟1) 𝑝(𝑟1) ⋅ ⋅ ⋅ 𝑇𝑒(𝑟2) 𝑇𝑖(𝑟2) 𝑁𝑒(𝑟2) 𝑝(𝑟2) ⋅ ⋅ ⋅ ... ... ... ... ... 𝑇𝑒(𝑟𝑁) 𝑇𝑖(𝑟𝑁) 𝑁𝑒(𝑟𝑁) 𝑝(𝑟𝑁) ⋅ ⋅ ⋅ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.20)

where 𝑟𝑖 represent diﬀerent grid points for 𝑖 = 1, ⋅ ⋅ ⋅ , 𝑁. 𝑇𝑒, 𝑇𝑖, 𝑁𝑒 and 𝑝 denote the electron

temperature, ion temperature, electron density, and composition, respectively. The method starts with initial estimates for all parameters at all altitudes. It then computes the theoretical ACF based on the parameters, imposes the range smearing (by taking into account the pulse shape), and makes a weighted comparison to the data, where weights are obtained from the data error covariances. The technique then proceeds by updating the parameter values so that the weighted diﬀerence between the actual data and the theoretical data is minimized.

The formulation above is in principle useful, but it is not the most eﬃcient way to perform the full-proﬁle analysis. The basic reason is that in order for the model to be accurate enough, the

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p

Figure 2.7 Flowchart of the eﬀective implementation of the full-proﬁle technique.

whose spacing is equal to the receiver sampling period. Unfortunately, considering such a dense grid increases the number of parameters of the optimization search space, which is rather costly in terms of computational power requirements, and it will be useful to develop more eﬃcient ways for analysis.

Attempts have been made to develop eﬀective implementation of the full-proﬁle technique based on a hierarchy of grids and interpolation methods between the grids [Lehtinen et al., 1996; Holt

et al., 1992]. The diﬀerence in interpolation methods makes the distinction among these various

implementations. Holt et al. [1992] suggest Spline interpolation whereas Lehtinen et al. [1996] use Lagrange and linear interpolations throughout a hierarchy of grids. Figure 2.7 represents the ﬂowchart of such methods.

Although performing the interpolation reduces the computational cost, the technique remains computationally expensive. One basic reason is due to the requirements of nonlinear computational techniques. One requirement is the computation of the derivative of the minimization function with respect to search variables (parameters in the coarse grid). Even though analytical expressions of

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the derivatives of the lags of the theoretical auto-correlation function with respect to the ionospheric parameters are available, they cannot be exploited in the optimization procedures of the full-proﬁle techniques. Therefore, forward diﬀerence, which slows down the speed of computation, is the only method that can be used for derivative calculation.

The full-profile technique is optimal in the sense that all the available information, such as the complete model of the ambiguity function and the full error covariance matrix, can be incorporated in the analysis. However, the computational cost of the method limits its routine implementation. In the next chapter, we develop the theory of a new hybrid inversion technique which aims at obtaining estimates that are close to optimal at a fraction of the usual computational cost. The technique is based on a correction to the effect of the transmitted waveform on the ACF lag profiles through a deconvolution process, and subsequent estimation of parameters from the plasma ACF at individual altitudes.

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CHAPTER 3 PROPOSED TECHNIQUE FOR INVERSION OF

INCOHERENT SCATTER MEASUREMENTS

The goal of this chapter is to introduce an efficient near-optimal technique for estimation of iono-spheric parameters from incoherent scatter measurements. The technique is based on a correction to the effect of the transmitted waveform on the ACF lag profiles through a deconvolution process, and subsequent estimation of parameters from the plasma ACF at individual altitudes. In this work, we focus on long-pulse measurements and investigate the performance of the hybrid tech-nique on simulated data (this chapter) and actual incoherent scatter radar measurements (Chapter 5).

In development of the hybrid technique, we exploit both the simplicity of height-by-height analysis and the accuracy of full-profile methods through considering the full model of ambiguity to present a simple, fast, and accurate method without the limitations of each of these common, currently used methods. For this purpose, we revisit the forward (direct) model of the incoherent scatter process as the form of the 1-dimensional convolution of the ionospheric ACF across range at each lag. We then present the matrix framework of convolution. The inversion technique is then formulated as the deconvolution of the lag profiles followed by the minimization of a least-squares cost function. Two different regularization methods for performing the deconvolution are discussed. Extension of the method to 2-dimensions is also discussed. The materials presented in Sections 3.1, 3.2, and 3.4.1 follow closely the description of the procedure reported by Nikoukar et al. [2008].

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3.1 Forward model

This section describes the discretization of the radar equation (Chapter 2) and the corresponding matrix framework of the problem. Let 𝑟′ ₌ 𝑐𝑡

2 − 𝑟 and rewrite the radar equation as

< 𝑣(𝑡)𝑣(𝑡 + 𝜏) > ∝ ∫ 𝑉0(2𝑟 ′ 𝑐 )𝑉0(𝜏 + 2𝑟′ 𝑐 )𝑅(k, 𝑐𝑡 2 − 𝑟′, 𝜏)𝑑𝑟′ (3.1) where 𝑐𝑡

2 refers to the altitude from which the signal is received (reference altitude). We discretize

and approximate both sides of the above equation by a Reimann sum, as below

< 𝑣(2𝑗Δ𝑟_𝑐 )𝑣(2𝑗Δ𝑟_𝑐 + 𝜏) > ∝ ∑∞

𝑖=−∞

𝑉0(2𝑖Δ𝑟_𝑐 )𝑉0(𝜏 +2𝑖Δ𝑟_𝑐 )𝑅(k, (𝑗 − 𝑖)Δ𝑟, 𝜏)Δ𝑟 ∀𝑗 (3.2)

where Δ𝑟 = sampling period₂ (𝛿𝑡)×𝑐, and 𝑖 and 𝑗 are indexing terms. Note that 𝑖 can range from

𝑗 − 𝑇 𝑐

2Δ𝑟 = 𝑗 − 𝛿𝑡𝑇 = 𝑗 − 𝑁𝑅 to 𝑗, where 𝑁𝑅 is the number of altitudes in each range-gate. This

discretization is necessary since eventually the data and the resolution of the ﬁnal parameter grid, Δ𝑟, are restricted by the sampling time of the receiver.

Following the notation introduced in Chapter 2, we replace 𝑉0(2𝑖Δ𝑟_𝑐 )𝑉0(𝜏 +2𝑖Δ𝑟_𝑐 ) in (3.2) by p𝜏(𝑖)

to obtain < 𝑣(2𝑗Δ𝑟_𝑐 )𝑣(2𝑗Δ𝑟_𝑐 + 𝜏) > ∝ 𝑗 ∑ 𝑖=𝑗−𝑁𝑅 p𝜏(𝑖)𝑅(k, (𝑗 − 𝑖)Δ𝑟, 𝜏) ∀𝑗 (3.3)

Note that the above equation describes the relationship between the input and output of a linear time-invariant system, where the plasma ACF at certain time lag, 𝜏, is the input, and the measured voltage ACF at the same time lag is the output. Moreover, the impulse response of the system is expressed as

p𝜏(𝑖) = 𝑉0(2𝑖Δ𝑟_𝑐 )𝑉0(𝜏 +2𝑖Δ𝑟_𝑐 ) for 𝑖 = 0, 1, ⋅ ⋅ ⋅ , 𝑁𝑅 and 𝜏 = 0, Δ𝜏, ⋅ ⋅ ⋅ , 𝑇 (3.4)

where Δ𝜏 represents the time lag increment. Notice that p𝜏(𝑖) preserves its form over diﬀerent

range-gates. Therefore, the index 𝑖 can be considered to vary over a range-gate only, rather than being dependent on particular altitude 𝑗.

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The above scheme can be visualized as incorporating the effects introduced by the transmitted pulse into a number of low-pass filters, each of which affects only one lag profile. These effects take place in the form of weighted averaging of the lag profiles, where the weights are determined by the ambiguity function at each time lag 𝜏. Thus, the filter shape at each lag in the time domain is determined by the product of the pulse shape and its shifted version.

Once we envision range smearing as a ﬁltering system, we can describe the relationship between its input and output, i.e. the plasma ACF at individual altitudes and the measured voltages ACF, as a convolution process; that is,

< 𝑣(2𝑖Δ𝑟_𝑐 )𝑣(2𝑖Δ𝑟_𝑐 + 𝜏) > = p𝜏(𝑖) ∗ 𝑅(k, 𝑖Δ𝑟, 𝜏) for ∀𝑖 and 𝜏 = 0, Δ𝜏, ⋅ ⋅ ⋅ , 𝑇 (3.5)

where * represents the convolution operation. With expansion of (3.5) with respect to all possible values of 𝑡 and ﬁxed value of 𝜏, we can represent this convolution relationship in a matrix framework as y𝜏 = P𝜏m𝜏 for ∀𝜏 (3.6) where y𝜏 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑣(𝑖2Δ𝑟 𝑐 )𝑣(𝑖2Δ𝑟𝑐 + 𝜏)∣𝑖=𝑁𝑅 𝑣(𝑖2Δ𝑟 𝑐 )𝑣(𝑖2Δ𝑟𝑐 + 𝜏)∣𝑖=𝑁𝑅+1 ... 𝑣(𝑖2Δ𝑟 𝑐 )𝑣(𝑖2Δ𝑟𝑐 + 𝜏)∣𝑖=𝑛 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , m𝜏 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑅(k, 𝑖Δ𝑟, 𝜏)∣𝑖=1 𝑅(k, 𝑖Δ𝑟, 𝜏)∣𝑖=2 ... 𝑅(k, 𝑖Δ𝑟, 𝜏)∣𝑖=𝑛 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ P𝜏 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ p𝜏(𝑖)∣𝑖=𝑁𝑅 ⋅ ⋅ ⋅ p𝜏(𝑖)∣𝑖=1 𝑛−𝑁_z}|{𝑅 0 0 p𝜏(𝑖)∣𝑖=𝑁𝑅 ⋅ ⋅ ⋅ p𝜏(𝑖)∣𝑖=1 𝑛−(𝑁𝑅+1) z}|{ 0 ... 𝑛−𝑁_z}|{𝑅 0 p𝜏(𝑖)∣𝑖=𝑁𝑅 ⋅ ⋅ ⋅ p𝜏(𝑖)∣𝑖=1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Notice that y𝜏 is the vector of measurement ACF at time lag 𝜏 from all altitudes (lag proﬁles).

Similarly, m𝜏 includes the altitude proﬁle (for 𝑛 altitudes) of the true plasma ACF at the same

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A more complete model of ISR measurement can be rewritten as

y𝜏 = P𝜏m𝜏+ 𝜖𝜏 ∀𝜏 (3.7)

where 𝜖𝜏 represents the measurement error at time lag 𝜏. In general this error is signal-dependent

especially when the backscattered signals are strong due to high electron densities or high transmit-ted power. Because of this dependance, it is not possible to obtain an estimate of the lag profiles using only one single transmission. Instead, the measured ACFs should be added for several pulse transmissions to improve statistical accuracy (integration). The data is typically integrated for several seconds. Although signal-dependent, the noise gains Gaussian characteristics as signal is integrated over many pulses (according to the central limit theorem). Therefore, it can be well described by its mean, 𝜇 =< 𝜖𝜏 >, and covariance matrix, Σ𝜖𝜏 =< 𝜖𝜏𝜖𝜏𝑇 >. The covariance matrix will be diagonal if the errors in different lags are independent. Otherwise it will have nonzero off-diagonal elements.

3.2 Inverse model

In the previous sections we established the relationship between the plasma ACF at individual altitudes and the received voltage ACF as a convolution process, where the shape of the convolving function is dependent on the pulse envelope as well as the time lag values. We exploit this property in our proposed inversion method, a detailed description of which is presented in the following sections.

3.2.1 Deconvolution

The major motivation for the deconvolution of the lag profiles is to remove the range smearing from the measured signal ACF and obtain the plasma ACF at single altitudes. The elimination of range ambiguity allows us to use analytical derivatives of the theoretical ACF lags with respect to ionospheric parameters, as opposed to forward differences, in least-squares optimization algorithms and, hence, to reduce the computational cost significantly. Furthermore, when using deconvolution methods, one does not require the imposition of unrealistic assumptions on the parameter profiles,

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such as stationarity, for the whole range-gate as is the case in height-by-height analysis.

Regularization

To perform the deconvolution task, we can use methods such as inverse filtering and least-squares analysis [Blahut, 2004; Lagendijk and Biemond, 2000; Karl, 2000]. These methods, although straightforward and easy to implement, suffer from a common drawback, which is the instabil-ity of the solution in the face of perturbations to data. This major drawback raises the need for regularization. Through regularization, we impose a priori knowledge about the underlying process to stabilize the solution in the presence of noise and to permit the identification of physically reason-able estimates of parameters of interest. A regularization method can be considered as a modified least-squares technique, where the modifications appear in the form of additional constraints to the residual norm defined in (3.8) as side constraint norms. More precisely, we can represent the regularized estimate as the solution to the following minimization problem

ˆ m𝜏,𝑟𝑒𝑔= arg min_m 𝜏 ( ∣∣y𝜏− P𝜏m𝜏∣∣2_Σ−1 𝜖𝜏 + ∑ 𝑖 𝜆𝑖𝐶𝑖(m𝜏) ) ∀𝜏 (3.8)

where 𝜆𝑖 and 𝐶𝑖 are the 𝑖th regularization parameter and regularization functional, respectively.

The first term controls data fidelity (i.e. how closely the solution fits the data), whereas the second term (the regularization term) controls how well the solution matches our prior knowledge. The role of the regularization parameter can be viewed as controlling the trade-off between the impact of data and the impact of a priori knowledge on the solution.

In what follows we introduce two methods of regularization which use diﬀerent side functionals.

A: Tikhonov regularization The most common regularization method is the Tikhonov regu-larization with a quadratic functional [Karl, 2000; Demoment, 1989]. The general expression for the Tikhonov method is

ˆ m𝜏,𝑇 𝑖𝑘ℎ𝑜𝑛𝑜𝑣= arg min_m 𝜏 ( ∣∣y𝜏 − P𝜏m𝜏∣∣2_Σ−1 𝜖𝜏 + 𝜆 2_∣∣Lm 𝜏∣∣2 ) ∀𝜏 (3.9)

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enforces a roughness penalty and, hence, a smoothness constraint. As an example, the discretized ﬁrst-order gradient operator can be represented as

L = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 −1 0 ⋅ ⋅ ⋅ 0 1 −1 0 ⋅ ⋅ ⋅ ... ... ... ... ... 0 ⋅ ⋅ ⋅ 1 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.10)

Note that ∣∣z∣∣𝑝𝑝 = (∑_𝑖𝑧𝑝_𝑖)1/𝑝. In this case ∣∣Lm𝜏∣∣2 is a measure of the variability of the estimate.

Therefore, the overall functionality of the method can be visualized as penalizing large gradients of the plasma ACF lag proﬁles, resulting in smoother lag proﬁles where the degree of smoothness depends on the value of the regularization parameter.

The solution to the minimization in (3.8), i.e. the Tikhonov regularized estimate, can be obtained as the solution to the following set of equations

(P𝑇_𝜏Σ−1_𝜖_𝜏 P𝜏+ 𝜆2L𝑇L) ˆm𝜏 = P𝑇𝜏Σ−1𝜖𝜏 y𝜏 ∀𝜏 (3.11)

B: Total variation (TV) regularization One drawback associated with Tikhonov regular-ization is that it severely penalizes the sharp gradients in ACF lag proﬁles; therefore, if there is a natural sharp gradient in the electron density proﬁle, for example, it will not be recovered by the Tikhonov method.

Total variation regularization is a nonlinear technique that tries to preserve sharp gradients in proﬁles [Karl, 2000; Vogel and Oman, 1996]. The general expression for the TV method is

ˆ m𝜏,𝑇 𝑉 = arg min_m 𝜏 ( ∣∣y𝜏 − P𝜏m𝜏∣∣2_Σ−1 𝜖𝜏 + ∣∣Lm𝜏∣∣1 ) ∀𝜏 (3.12)

The ℓ1norm used in the TV technique does not penalize the sharp edges in lag proﬁles as severely

as the quadratic norm used in the Tikhonov method. Thus it is well suited for situations where the plasma parameter altitude proﬁles contain sharp gradients.

Near-optimal inversion of incoherent scatter radar measurements: coding schemes, processing techniques, and experiments

ABSTRACT

ACKNOWLEDGMENTS

TABLE OF CONTENTS

CHAPTER 1

INTRODUCTION